For Ring Ring (hanicim) and Ring Ring Ring (kezirin): nosebabI liked this puzzle. I used trial-and-error instead of a theory-based approach. Although I guess I was thinking that the machine calculates f(x) = x * (0 - x) and that would have to have a fixed-point (other than zero). Clearly f(-1) = -1 no matter what modulus you use but I don't have any proof that some of the rest of the odd numbers wouldn't fall into a separate cycle. Luckily things worked out pretty well.